309: Linear Analysis Spring 2016

Lecture 14: April 29

Lecturer: Lucas Van Meter

14.1 Euler-Fourier Formulas

Last time we considered the family of functions

1, sin(nπx/L), and cos(nπx/L) where n = 1, 2, 3, . . .

The big idea is that these functions form an orthogonal basis for the vector space of continuous functionswith period 2L. Because they form a basis if f (x) is any continuous function with period 2L we should beable to write ∞ a0 X h nπx πx i f (x) = + an cos + bn sin n 2 n=1 L L

In conclusion, for this wave function

∞ 1 X 2 f (x) = − sin ((2n + 1)πx) . 2 n=0 (2n + 1)π

14.1.1 More examples

Example 14.2 Consider the function

f (x) = ex , −L < x < L, with f (x + 2L) = f (x) for all x.What is the Fourier series?Let’s sketch this function before going on.First we note that this function does have period 2L so it should have a Fourier series. We use the Euler-Fourier formulas to find the coefficients.

Before the next example let’s recall the notion of even and odd functions. A function is even if f (−x) = f (x)for all x. A function is odd if f (−x) = −f (x). Examples of even functions are 1, x2 , x4 , x2 − 2x6 , cos(x).Examples of odd functions are x, x3 , x5 − 2x1 1, sin(x).Notice that even and oddness are preserved by linear combinations and we can even say something aboutthe product of even and odd functions.Notice that if f (x) is odd then Z L f (x)dx = 0 −Land if f (x) is even then Z L Z L f (x)dx = 2 f (x)dx. −L 0

The main use we will have is that if a function is odd then its Fourier series will only contain sine terms. Ifa function is even it will only contain constant and cosine terms.Lecture 14: April 29 14-3